#8211700

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#54835	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(34,36)(35,41)(40,44)(43,47);(3,9)(7,10)(11,13)(15,16)(17,19,23,20)(18,25,22,24)(21,27,30,28)(26,32,29,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8118896);